An attempt at showing that function is lipschitzian Let $X$ be normed space and let $B$ denote the closed unit ball in $X$. Suppose we have map $T_0: B \rightarrow B$ which satisfies Lipschitz condition with some constant $k > 1$. We extend this map to $T_1: 2B \rightarrow B$ in the following way:
$$ T_1x = \begin{cases} T_0x, & \text{for } \|x\| \leq 1, \\
(2-\|x\|)T_0(Px), & \text{for } 1 < \|x\| \leq 2,
\end{cases}$$
where $Px = \frac{x}{\|x\|}$ for $\|x \| > 1$ and $Px = x$ for $\|x\| \leq 1$ ($P$ is radial projection onto unit ball). Im reading a paper where the author claims this extended map still satisfies Lipschitz condition, this time with constant $2k + 1$. I tried to verify this.
$\textbf{Case 1}:$ If $x,y \in B$, then $\|T_1x - T_1y\| = \|T_0x - T_0y\| \leq k\|x - y\|$.
$\textbf{Case 2}:$ Let $x \in B$ and $y \in 2B \setminus B$. Then
\begin{equation} 
\begin{split}
\|T_1x - T_1y\| & = \|T_0x - (2-\|y\|)T_0(Py)\| \\ 
&\leq \Bigl\|T_0x - T_0(Py) - T_0(Py) + \|y\|T_0(Py) \Bigr\| \\
&\leq \|T_0(Px) - T_0(Py)\| + (\|y\| - 1)\|T_0(Py)\| \\
&\leq k\|Px - Py\| + \|y\| - \|x\| \\ 
&\leq 2k\|x - y\| + \|x - y\| \\
& = (2k + 1)\|x - y\|.
\end{split}
\end{equation}
Here I used the fact that in every normed space projection $P$ is Lipschitz with constant $2$.
$\textbf{Case 3}:$ Now this is where I can't finish the job. Let $x,y \in 2B \setminus B$. I only managed to show that this map is Lipschitz with constant $8k + 1$.
\begin{equation}
\begin{split}
\|T_1x - T_1y\| & = \Bigl\|(2-\|x\|)T_0(Px) - (2-\|y\|)T_0(Py) \Bigr\| \\ 
& = \Bigl\|2T_0(Px) - 2T_0(Py) - \|x\|T_0(Px) + \|y\|T_0(Py) \Bigr\| \\
& \leq 2\|T_0(Px) - T_0(Py)\| + \Bigl\| \|x\|T_0(Px) - \|y\|T_0(Px) \Bigr\| + \Bigl \|\|y\|T_0(P_x) - \|y\|T_0(Py) \Bigr\| \\
& \leq 4k\|x - y\| + \Bigl | \|x\| - \|y\| \Bigr | \|T_0(Px)\| + \|y\|\|T_0(Px) - T_0(Py)\| \\
& \leq 4k\|x - y\| + \|x-y\| + 4k\|x-y\| \\
& = (8k + 1)\|x - y\|.
\end{split}
\end{equation}
This is all I can get at the moment. Any help will be appreciated.
$\textbf{Edit}:$ There is extra assumption on map $T_0$ which I didnt mention, because I thought its irrelevant in evaluating Lipschitz constant. Namely, we assume $\inf\{\|x - T_0x\|: x \in B\} > 0$ (such maps exist in infinite dimensional spaces).
 A: Suppose, without loss of generality, that $\left\Vert x\right\Vert >\left\Vert y\right\Vert$.
Then we have $$
\begin{align}
\left\Vert T_{1}\left(x\right)-T_{1}\left(y\right)\right\Vert 
& =
\left\Vert \left(2-\left\Vert x\right\Vert \right)T_{0}\left(Px\right)-\left(2-\left\Vert y\right\Vert \right)T_{0}\left(Py\right)\right\Vert 
\\ & = 
\left\Vert \left(2-\left\Vert x\right\Vert \right)T_{0}\left(Px\right)-\left(2-\left\Vert x\right\Vert +\left\Vert x\right\Vert -\left\Vert y\right\Vert \right)T_{0}\left(Py\right)\right\Vert 
\\ & = 
\left\Vert \left(2-\left\Vert x\right\Vert \right)T_{0}\left(Px\right)-\left(2-\left\Vert x\right\Vert \right)T_{0}\left(Py\right)-\left(\left\Vert x\right\Vert -\left\Vert y\right\Vert \right)T_{0}\left(Py\right)\right\Vert 
\\ & \leq 
\left\Vert \left(2-\left\Vert x\right\Vert \right)T_{0}\left(Px\right)-\left(2-\left\Vert x\right\Vert \right)T_{0}\left(Py\right)\right\Vert +\left\Vert \left(\left\Vert x\right\Vert -\left\Vert y\right\Vert \right)T_{0}\left(Py\right)\right\Vert 
\\ & =
\left(2-\left\Vert x\right\Vert \right)\left\Vert T_{0}\left(Px\right)-T_{0}\left(Py\right)\right\Vert +\left(\left\Vert x\right\Vert -\left\Vert y\right\Vert \right)\left\Vert T_{0}\left(Py\right)\right\Vert 
\end{align}
$$
for $x,y \in 2B/B$.
Since
$\left\Vert x\right\Vert >\left\Vert y\right\Vert$
we have
$\left\Vert x\right\Vert -\left\Vert y\right\Vert =\left|\left\Vert x\right\Vert -\left\Vert y\right\Vert \right|\leq\left\Vert x-y\right\Vert $
, since
$ 1 < \left\Vert x\right\Vert $
we have
$ 2-\left\Vert x\right\Vert < 1 $
and since
$ T_{0}\left(Py\right)\in B $
we have
$ \left\Vert T_{0}\left(Py\right)\right\Vert \leq 1 $.
Therefore
$$
\begin{align}
\left(2-\left\Vert x\right\Vert \right)\left\Vert T_{0}\left(Px\right)-T_{0}\left(Py\right)\right\Vert +\left(\left\Vert x\right\Vert -\left\Vert y\right\Vert \right)\left\Vert T_{0}\left(Py\right)\right\Vert 
& <
\left\Vert T_{0}\left(Px\right)-T_{0}\left(Py\right)\right\Vert +\left\Vert x-y\right\Vert
\\ & \leq 
k\left\Vert Px-Py\right\Vert +\left\Vert x-y\right\Vert 
\\ & \leq 
2k\left\Vert x-y\right\Vert +\left\Vert x-y\right\Vert 
\\ & =
\left(2k+1\right)\left\Vert x-y\right\Vert 
\end{align}
$$
as claimed.
